Structural, mechanical and electronic properties of Nb₂C: first-principles calculations

Abstract

Nb–C compounds are potential candidates to achieve high hardness and refractory nature. We performed a crystal structure search for the Nb–C system using an ab initio evolutionary algorithm implemented in the USPEX code. By comparing the formation enthalpy, a P–x phase diagram was calculated, and an orthorhombic Pnnm structure of Nb₂C was predicted and denoted as Nb₂C-I, which was both mechanically and dynamically stable. In this Pnnm phase, there are four-sided rings continuously along the c-axis, which probably contributes to the relatively high incompressibility of Nb₂C-I along the crystallographic c-axis. Moreover, the hardness and Young's modulus were calculated to be 28.5 GPa and 448.9 GPa, respectively; hence, the Nb₂C-I is considered as a potential ultra-stiff and hard material.

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1 Introduction

In the past few decades, superhard materials with excellent mechanical properties have drawn extensive attention from the researchers and have been well-studied in terms of both theoretical and experimental aspects.¹ Traditional superhard materials, such as diamond,^2,3 γ-B,⁴ c-BN,⁵ and c-BC₂N,^6,7 are mainly composed of light elements. Diamond, the earliest discovered and hardest (70–150 GPa) superhard material, shows promising mechanical properties and is hence among the most widely used superhard materials. However, diamond shows its own practical defects. At a temperature greater than 800 °C, diamond tends to oxidize. Moreover, when the diamond tool is used in the grinding or cutting of ferrous materials, the tool easily reacts with iron-based materials. The c-BN, another typical superhard material, is difficult to synthesise, which limits its practical application. Therefore, development of novel superhard multifunctional materials is an urgent requirement.

During the past decade, scientists were focused on the design of new materials in transition-metal (TM) light-element (LE) systems. One of the advantages of transition metals is their large bulk modulus. For example, the bulk modulus of Os is about 395–462 GPa,⁸ which is close to that of diamond (446 GPa). However, the hardness of metal is always low, which is normally one-thirtieth that of diamond. This is mainly due to the huge difference of chemical bonds in the compound structures. Classical superhard materials usually hold three-dimensional networks composed of covalent bonds with sp³ hybridization, which is the basis of superhard properties. Therefore, the type of TM-LE bonding is the key factor for improving the hardness. Due to the behavior of high valence electron density and large interatomic distance, the transition metal can be intercalated with light elements, which may lead to the formation of covalent bonding between light elements and transition metals. Thus, the relatively high directional covalent bonds and valence electron density are considered to favor the formation of new superhard materials. It is believed that the carbides of the group-IV, -V, and -VI transition metals show an unusual combination of physical and chemical properties.^9–14 The transition-metal carbide (TMC) compounds combine the physical properties of three different kinds of materials: transition metals, covalent solids and ionic crystals,¹⁵ making TMC a family of industrially relevant compounds with outstanding physical properties.

The niobium carbides are an emblematical TMC system, which exhibits meaningful physicochemical properties.^10,16 Moreover, niobium carbides are significant candidates for improving the mechanical properties of niobium alloys.^10,16–19 Niobium and carbide could form weak metallic or semiconductors compounds with different compositions and space symmetries. In the Nb–C binary phase diagram, niobium carbides have four experimentally known compositions: NbC, Nb₆C₅, Nb₄C₃, and Nb₂C.^20–22 Stoichiometric NbC is in the rocksalt structure,²³ in which Nb atoms form the cubic lattices, and C atoms occupy the octahedral interstitial positions. Nb₄C₃ was reported to adopt a R3̄m space group, and Nb₆C₅ is in the P3₁ space group.^24,25 Among all the niobium carbides, Nb₂C is the least understood carbide;²⁶ it exists in three polymorphic forms: γ-Nb₂C, β-Nb₂C, and α-Nb₂C,^27–31 as reported in previous studies. The γ- and β-Nb₂C are hexagonal phases. The γ-Nb₂C is in L'3-type structure at temperatures above 2770 K. The β-Nb₂C has ζ-Fe₂N structure at relatively low temperatures. The α-Nb₂C is an orthorhombic low temperature phase. As Hugosson reported,²⁰ some controversy exists in the determination of the structure of the orthorhombic phase. In his study, both structures showed energies of formation higher than those of the hexagonal phase. However, other studies showed that the formation energy of α-Nb₂C was the lowest, which indicated that the formation of α-Nb₂C was feasible.^26,32 To date, controversy still remains regarding the orthorhombic phase structure of Nb₂C.

In this article, we performed a crystal structure search for the Nb–C system using an ab initio evolutionary algorithm implemented in the USPEX code. Full P–x phase diagrams of the Nb–C system at 0 K were calculated. We report a new phase for Nb₂C at high pressures. The predicted new phase of Nb₂C belongs to the orthorhombic Pnnm space group. Moreover, this phase is dynamically and mechanically stable at ambient and high pressures. Through a detailed electronic structure analysis, we found that niobium and carbon formed strong bonds with sp³ hybridization style, which further knitted into three-dimensional networks. Collectively, we proved that Nb₂C can be a potential candidate for an ultra-stiff and hard material.

2 Methods

To search the stable and low-enthalpy structures of the Nb–C system, the evolutionary algorithm USPEX^33–35 coupled with an ab initio evolutionary algorithm (EA) was employed. The calculation was designed to find the most stable structure for the given stoichiometries under given external conditions as implemented in the USPEX code. In this study, evolutionary crystal structure prediction calculations were performed at 0, 10, 30, and 50 GPa for the Nb_xC_y system (x = 1–8, y = 1–8) at zero temperature. We performed ab initio calculations with the local density approximation (LDA), as implemented in the Vienna ab initio simulation package (VASP) code,^36,37 which is based on density functional theory. The electron–ion interaction was described via the projector augmented wave (PAW) scheme.³⁸ The orbitals 2s and 2p for C and 4p, 4d, and 5s for Nb were treated as valence states. During structural relaxation, a tested energy cutoff of 600 eV was used for the plane wave basis sets, and Monkhorst–Pack³⁹ k-point meshes with a resolution of 0.03 Å⁻¹ in the reciprocal space were used for all structures to minimize error from the k-point meshes. The atomic positions, lattice parameters, and cell volume were fully relaxed until the force on each atom was less than 1 meV Å⁻¹. Phonons were calculated using the supercell method implemented in the PHONOPY package.^40,41 The strain–stress method was used to calculate the elastic constants. The shear modulus, bulk modulus, Young's modulus, and Pugh's indicator were derived from the Voigt–Reuss–Hill approximation.^42–44 The structure diagram of Nb₂C was drawn using VESTA.⁴⁵

3 Results and discussion

In this study, we uncovered a stable structure of Nb₂C, denoted as Nb₂C-I. At first, we calculated the formation enthalpy of Nb_xC_1−x using a fractional representation of Nb_xC_1−x with respect to its decomposition into Nb and C as

ΔH(Nb_xC_1−x) = H(Nb_xC_1−x) − [x × H(Nb) + (1 – x) × H(C)]

where x is the concentration of Nb. The formation enthalpies from 0 to 50 GPa were evaluated as the difference in the enthalpy of the predicted Nb–C system with respect to Nb metal and C in their most stable forms (bcc phase for Nb and graphite for C), as depicted in Fig. 1. The formation enthalpy of NbC (Fm3m) in our calculation is −0.531 eV, similar to the previous results²⁰ (−43 mRy = −0.585 eV). Structures lying on the convex hull are either thermodynamically stable or metastable and can be synthesized in principle. According to Fig. 1(a), at 0 GPa, the Nb₂C-Pnma and Nb₆C₅ phases are thermodynamically stable, and Nb₄C₃ is also located on the convex hull, which is similar to the previous results.³² With an increase in the pressure, the stoichiometric NbC phase becomes thermodynamically stable, and Nb₂C-Pnnm is more stable than Nb₂C-Pnma at 30 GPa. We calculated the formation enthalpies of three candidates of Nb₂C. As Fig. 2 shows, the enthalpy curves of Nb₂C structures are presented. It can be clearly seen that the Nb₂C-Pnma is the most stable structure at 0 GPa, which is consistent with the Wu's studies.²² When the pressure is greater than 15.8 GPa, the formation enthalpies of the proposed Nb₂C-I are lowest negative values, which indicates that Nb₂C-I is thermodynamically stable under pressure. From Fig. 1(d), it can be observed that at 50 GPa, the stable phases are only NbC and Nb₆C₅. We calculated the detailed enthalpy differences as a function of pressure. As shown in the ESI Fig. S1(a),† the enthalpy difference of NbC is calculated against decomposition into Nb₆C₅ and C; thus, we concluded that the NbC phase becomes stable at 12.3 GPa. The enthalpy differences of Nb₄C₃ and Nb₂C-Pnnm are calculated against decomposition into the constituent carbides. As shown in Fig. S1(b) and (c),† the structures of Nb₄C₃ and Nb₂C-Pnnm become completely unstable at 39.7 GPa and 40.5 GPa, respectively. With these detailed calculations shown in Fig. S1,† a pressure–constituent (P–x) phase diagram of niobium carbide was constructed, which is plotted in Fig. 3. With the increase in C concentration, the Nb–C system first undergoes an orthorhombic to monoclinic transition and then transforms back to the cubic phase. Moreover, some phases of niobium carbides appear and decompose with the increase of pressure. However, the structure with high carbon concentration is not stable, even if the pressure is increased to 50 GPa.

Fig. 1 Convex hull diagram for the Nb–C compounds. The formation enthalpies (ΔH, with respect to Nb and C of their most stable phases) of Nb_xC_1−x. The abscissa x is the fraction of Nb in the structures. Circles on the solid lines represent stable ground-state compounds.

Fig. 2 Enthalpy–pressure diagrams. Calculated enthalpies as the function of pressure.

Fig. 3 Pressure–composition phase diagrams.

The Pnnm structure, as we predicted, contains two Nb₂C in a unit cell at 20 GPa (a = 5.415 Å, b = 4.837 Å, and c = 3.019 Å), in which two inequivalent atoms Nb and C occupy the Wyckoff 8h (0.3462, 0.7632, 0.5000) and 2b (0.5, 0.5, 0.0) sites, as summarized in Table 1 and shown in Fig. 4. Fig. 4(b) along the y-axis and Fig. 4(c) along the z-axis reveal a fundamental structure in the Pnnm phase. Fig. 4(d) shows the structure of Pnma phase along the y-axis. When we compared Fig. 4(b) with (d), it was observed that the carbon atoms in Nb₂C-Pnnm could help to achieve better formation of the three-dimensional space grid structure, which avoided the appearance of the interaction between metal atoms.

Table 1 The optimized equilibrium lattice constants a, b, and c (Å), and atomic coordinates for Nb₂C-I at 20 GPa

	Structure	Parameters (Å)	Atom	x	y	z
Nb2C	Pnnm (orthorhombic)	a = 5.415, b = 4.837, c = 3.019	Nb(8h)	0.3462	0.7632	0.5000
			C(2b)	0.5000	0.5000	0.0000

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Fig. 4 Crystal structures of the predicted Nb₂C-Pnnm. (a) The structure of Nb₂C-Pnnm. (b) The structure along the y-axis. (c) Along the z-axis. The large gray and small blue spheres represent the Nb and C atoms, respectively. (d) The structure of Nb₂C-Pnma along the y-axis.

It is well-known that the phonon dispersion curves give information about the global structural stability of the materials. As shown in Fig. 5, we analyzed the phonon dispersion curves to test the lattice dynamical stability of Nb₂C-I at 0 and 20 GPa. The calculated phonon curves of Nb₂C have no soft mode in the Brillouin zone, indicating that Nb₂C is dynamically stable. For the purpose of understanding the mechanical properties, the elastic constants are deemed essential. The elastic constants C_ij of the Pnnm phases are listed in Table 2. For the proposed Nb₂C-Pnnm, all C_ij satisfy Born–Huang criteria,⁴⁶ which means that the proposed Nb₂C-Pnnm is mechanically stable. We determined that the Nb₂C-Pnnm could be synthesized under high pressure and preserved under ambient pressure. From Table 2, it can be found that the calculated C₃₃ value is bigger than the values of C₁₁ and C₂₂, indicating that there is relatively high incompressibility along the c-axis. The relative high incompressibility of the proposed Nb₂C along the c-axis is perhaps contributed by the existence of four-sided rings, which are continuously along the crystallographic c-axis. In contrast, four-sided rings and eight-sided rings exist alternately along the b-axis or c-axis. Therefore, there is certain disparity between C₃₃ and C₁₁ (or C₂₂).

Fig. 5 Dynamic stability of Nb₂C-I. The phonon dispersion curves of Nb₂C-I along the high symmetry directions of the Brillouin zone at 0 and 20 GPa.

Table 2 Elastic constants C_ij, Poisson's ratio ν, bulk modulus B (GPa), shear modulus G (GPa), Pugh's indicator G/B, Young's modulus E (GPa), and Vickers hardness H_V of Nb₂C-Pnma and Nb₂C-Pnnm at 0 GPa

	C11	C22	C33	C44	C55	C66	C12	C13	C23
Pnma	390	405	390	76	119	122	157	129	136
Pnnm	383	389	415	101	102	137	156	153	127

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	ν	B	G	G/B	E	HV
Pnma	0.19	236.1	178.1	0.75	426.9	26.8
Pnnm	0.19	244.5	188.0	0.77	448.9	28.5

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It is well-known that superhard materials should have high bulk modulus and high shear modulus to resist the volume change and shape change. At 0 and 20 GPa, the bulk modulus of Nb₂C-Pnnm is 244 and 310 GPa, respectively, which are larger than the values of Nb₂C-Pnma (236 and 304 GPa, respectively). Because the value of the bulk modulus is large, Nb₂C can be grouped into incompressible materials. As reported in Table 2, the Poisson's ratio ν of Nb₂C-Pnnm is about 0.19. Usually, strong directionality degree of covalent bonding is considered in the materials when the ν value is near 0.2, indicating that the directionality degree of covalent bonding of Nb₂C-Pnnm is strong. The Pugh's indicator G/B of Nb₂C is 0.77, which is a relatively large value. According to the Pugh's modulus ratio defined by Cheng et al.,⁴⁷ the Nb₂C-I phase is a brittle and hard material with a huge capability to resist elastic plastic deformation. To gain a more comprehensive and profound understanding of the mechanical property, we calculated the Vickers hardness of Nb₂C-I. The Vickers hardness H_V, estimated by the empirical model, was obtained by the following formula:

H_V = (k²G)^0.585 − 3 (k = G/B)

The calculated hardness of Nb₂C-I is 28.5 GPa, which almost matches that of the hard material WC (21.5–33.4 GPa).^47,48 Our results suggested that the proposed Nb₂C-I could be a potential candidate for ultra-stiff and hard materials.

The electronic structure is crucial to understand the origin of physical properties of these carbides. The total and site projected electronic densities of states (PDOS) of Nb₂C-I are shown in Fig. 6(a). As the graphic shows, Nb₂C is a metal as the d electronics of Nb are mainly attributed to the density near the Fermi energy in our calculation model. To determine the hybridization between C and Nb, projected electronic densities of states of the nearest Nb and C are calculated, as shown in Fig. 6(b). There is an obvious hybridization between C 2p and Nb 4d states, which is a common feature of typical TMC superhard materials. To obtain more information about the bonding character, the electronic localization function (ELF) of Nb₂C-I was calculated, as shown in Fig. 7. The electron localization function between Nb atoms and C atoms has a slight increase as compared to that in the background, which indicates that mixed bonds between the two atoms are present in this area, in which the ionic bonds are dominant. As Becke defined,⁴⁹ the electrons can move freely in the areas of ELF = 0.5. In Fig. 7(a), the green region is almost connected. However, there are disconnected equivalent spheres in Fig. 7(b). In total, the equivalent sphere of ELF = 0.5 is partially connected in the whole crystal, which may enhance the weak metallicity properties.

Fig. 6 Electronic density of states of Nb₂C-I. The DOS (a) and PDOS (b) of Nb₂C-I. The vertical black dashed line denotes the Fermi level.

Fig. 7 Contours of the electronic localization functions (ELF). Electron localization function isosurface maps for (a) Miller indices: (1.67 1 0) and (b) Miller indices: (1 0 0).

4 Summary

In summary, we researched the full-scale zero-temperature Nb–C phase diagram using an ab initio evolutionary algorithm implemented in the USPEX code. The P–x phase diagrams of the Nb–C system are constructed. A new structure of Nb₂C has been uncovered and denoted as Nb₂C-I. The Nb₂C-I belongs to the orthorhombic system with the space group Pnnm. Phase stability, mechanical properties, and electronic properties of Nb₂C-I were investigated. The Vickers hardness and Young's modulus of Nb₂C-I have been calculated to be 28.5 GPa and 448.9 GPa, respectively, which prove that Nb₂C-I can be considered as low compressible materials. We considered that the formation of continuous four-sided rings along the c-axis might cause high incompressibility of Nb₂C-I in the c-axis direction. A deep analysis of the electronic density of states and chemical bonding indicates that an ionic bond is dominant in the Nb₂C crystals.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (No. 2016YFB0700505, 2016YFB0701401). Parts of the results described in this paper are obtained on the Era of Computer Network Information Center of Chinese Academy of Sciences.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7ra05856j

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